Problem 48 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(13-48\) . $$ \int 3 x^{5} \sqrt{x^{3}+1} d x $$

Step-by-Step Solution

Answer

\( \frac{2}{5}(x^3 + 1)^{5/2} - \frac{2}{3}(x^3 + 1)^{3/2} + C \)

Step 1: Use Substitution

Let's make the substitution to simplify the integral. Set \( u = x^3 + 1 \). Then, the derivative \( du = 3x^2 \, dx \). Thus, we can express \( dx \) as \( dx = \frac{du}{3x^2} \). In our integral, we have \( 3x^5 \), which breaks down into \( 3x^3 \cdot x^2 \). Knowing \( u = x^3 + 1 \Rightarrow x^3 = u - 1 \), substitute into the integral.

Step 2: Substitute into the Integral

Substitute \( u = x^3 + 1 \) and \( dx = \frac{du}{3x^2} \) into the expression:\[\int 3x^5 \sqrt{x^3 + 1} \, dx = \int 3(x^3)(x^2) \sqrt{u} \frac{du}{3x^2} = \int (u - 1) \sqrt{u} \, du\]the \( x^2 \) term cancels out.

Step 3: Simplify the Expression

Now simplify the expression:\[\int (u^{3/2} - u^{1/2}) \, du\]

Step 4: Integrate Term by Term

Integrate each term separately:\[\int u^{3/2} \, du = \frac{u^{5/2}}{5/2} = \frac{2}{5}u^{5/2}\]\[\int u^{1/2} \, du = \frac{u^{3/2}}{3/2} = \frac{2}{3}u^{3/2}\]Now, combine these results:\[\frac{2}{5}u^{5/2} - \frac{2}{3}u^{3/2} + C\]

Step 5: Substitute Back in Terms of x

Substitute \( u = x^3 + 1 \) back into the equation:\[\frac{2}{5}(x^3 + 1)^{5/2} - \frac{2}{3}(x^3 + 1)^{3/2} + C\]Thus, the final result for the integral is:

Key Concepts

Integration by Substitution

Integration by substitution is a method used to simplify a given integral, making it easier to solve. This method is especially helpful when dealing with complicated algebraic expressions or when the integrand can be expressed as a function of another function. To effectively use this technique, follow these steps:

• Select a part of the integrand to substitute with a new variable, usually denoted as u. This substitution should simplify the integral.
• Find the derivative of u with respect to x, denoted as du/dx, and express dx in terms of du.
• Substitute all instances of x in the original integral with the new variable u and express dx accordingly. Your goal is to eliminate all x terms from the integral.
• Once the substitution is complete, perform the integration with respect to u.
• Finally, revert your substitution by replacing u back with the original x expression. This yields the integral in terms of x.

In the example provided, we substitute u = x^3 + 1. This clever choice simplifies the expression, enabling us to reframe the entire integral in terms of u, which is easier to solve.

Indefinite Integrals

Indefinite integrals are natively expressions that refer to the general form of the antiderivative of a function. They include a constant of integration, usually denoted as C. The importance of indefinite integrals is that they offer the family of functions that could describe the original function before differentiation. Here's how to evaluate indefinite integrals:

• Identify the function you wish to integrate, called the integrand.
• Apply the rules of integration, such as the power rule, which states that \[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\] where n is not equal to -1.
• Remember, because indefinite integrals include the constant C, your solution should reflect the potential shifts up or down the y-axis (vertical shifts) of the graph of the integrated function.
• For more complicated expressions, employ techniques such as integration by substitution to simplify the process.

The original exercise involves integrating powers of u, where each power term is integrated following the power rule, resulting in a new power and a constant of integration.

Algebraic Manipulation

Algebraic manipulation is a key skill necessary for performing integration and transforming expressions into forms that are easier to handle. This involves performing operations such as factoring, expanding, simplifying, or substituting expressions. For successful algebraic manipulation:

• Identify and factor expressions wherever possible to simplify the integrand.
• Break down complex expressions into simpler parts that can be recombined after integration.
• Use substitution effectively to reduce compound expressions into simpler ones.
• Cancel out terms when appropriate to further simplify the integrand.

In this exercise, algebraic manipulation allowed the integral \[3x^5 \sqrt{x^3 + 1} \] to transform by expressing the powers of x in terms of u. This leads to a simpler integral expression like \[(u - 1) \sqrt{u} \].After simplification, it becomes easier to integrate term by term, showcasing the power of both substitution and algebraic manipulation in calculus.